Download 7.3 - CSE

Section 7.3
Multiplying
and Simplifying
Radical
Expressions
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Objective #1
Use the product rule to multiply radicals.
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Multiplying Radicals
The Product Rule for Radicals
If n a and n b are real numbers, then
n
a ⋅ n b = n ab .
The product of two nth roots is the nth root of the product
of the radicands.
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Multiplying Radicals
EXAMPLE
4
Multiply: (a) 3 5 ⋅ 3 4 (b) 6 x − 5 ⋅ 6 (x − 5) .
SOLUTION
In each problem, the indices are the same. Thus, we multiply
the radicals by multiplying the radicands.
(a) 3 5 ⋅ 3 4 = 3 5 ⋅ 4 = 3 20
(b) 6 x − 5 ⋅ 6 ( x − 5) = 6 ( x − 5)( x − 5) = 6 ( x − 5)
4
4
5
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Objective #1: Example
1a. Multiply:
5 ⋅ 11
5 ⋅ 11 = 5 ⋅11
= 55
1b. Multiply:
x + 4 ⋅ x − 4=
=
x + 4⋅ x − 4
( x + 4)( x − 4)
x 2 − 16
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Objective #1: Example
1a. Multiply:
5 ⋅ 11
5 ⋅ 11 = 5 ⋅11
= 55
1b. Multiply:
x + 4 ⋅ x − 4=
=
x + 4⋅ x − 4
( x + 4)( x − 4)
x 2 − 16
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Objective #1: Example
1c. Multiply:
7
7
7
7
2 x ⋅ 6 x3
7
2 x ⋅ 6 x3 =
12 x 4
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Objective #1: Example
1c. Multiply:
7
7
7
7
2 x ⋅ 6 x3
7
2 x ⋅ 6 x3 =
12 x 4
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Objective #2
Use factoring and the product rule to simplify
radicals.
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Simplifying Radicals
Simplifying Radical Expressions by Factoring
A radical expression whose index is n is simplified when its radicand
has no factors that are perfect nth powers. To simplify, use the
following procedure:
1) Write the radicand as the product of two factors, one of which is
the greatest perfect nth power.
2) Use the product rule to take the nth root of each factor.
3) Find the nth root of the perfect nth power.
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Simplifying Radicals
EXAMPLE
2 3
Simplify by factoring: (a) 28 (b) 3 − 32 x y .
SOLUTION
(a) 28 = 4 ⋅ 7
= 4⋅ 7
=2 7
( )
(b) 3 − 32 x 2 y 3 = 3 − 8 y 3 4 x 2
4 is the greatest perfect square
that is a factor of 28.
Take the square root of each
factor.
Write 4 as 2.
− 8y 3 is the greatest perfect cube
that is a factor of the radicand.
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Simplifying Radicals
CONTINUED
= 3 − 8 y3 ⋅ 3 4x2
Factor into two radicals.
= −2 y 4 x
Take the cube root of − 8 y 3 .
3
2
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Simplifying Radicals
EXAMPLE
3
If f (x ) = 3 48(x − 2) , express the function, f, in simplified form.
SOLUTION
Begin by factoring the radicand. There is no GCF.
f (x ) = 3 48(x − 2 )
3
This is the given function.
= 3 6 ⋅ 8( x − 2 )
3
Factor 48.
= 6 ⋅ 2 (x − 2)
3
3
Rewrite 8 as 23.
3
= 3 6 ⋅ 3 23 ⋅ 3 ( x − 2 )
Take the cube root of each factor.
= 2( x − 2 ) 3 6
Take the cube root of 23 and
(x − 2)3 .
3
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Simplifying Radicals
Simplifying When Variables to Even Powers in
a Radicand Are Nonnegative Quantities
For any nonnegative real number a,
n
a n = a.
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Simplifying Radicals
EXAMPLE
Simplify:
40 x 3 .
SOLUTION
We write the radicand as the product of the greatest perfect
square factor and another factor. Because the index of the
radical is 2, variables that have exponents that are divisible by 2
are part of the perfect square factor. We use the greatest
exponents that are divisible by 2.
Use the greatest even power of
40 x 3 = 4 ⋅10 ⋅ x 2 ⋅ x
each variable.
=
(4 ⋅ x )(10 x )
2
= 4 x 2 ⋅ 10 x
Group the perfect square factors.
Factor into two radicals.
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Simplifying Radicals
CONTINUED
= 2 x 10 x
Simplify the first radical.
Remember – we take the principal root which is the positive root if
there is a choice between a negative and a positive. Odd roots of a
negative number are negative and we don’t have a choice. When we
take the even root of a positive number – we could argue that we
might get either a negative or a positive. But we take the positive, for
that is the principal root. Since we don’t know the sign of the variable
x in this problem and x could be either negative or positive – we put
absolute value bars around the x to assure that we have chosen the
positive root.
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Simplifying Radicals
EXAMPLE
Simplify:
4
96 x11 .
SOLUTION
We write the radicand as the product of the greatest 4th power
and another factor. Because the index is 4, variables that have
exponents that are divisible by 4 are part of the perfect 4th factor.
We use the greatest exponents that are divisible by 4.
4
96 x11 = 4 16 ⋅ 6 ⋅ x 8 ⋅ x 3
(
)( )
Identify perfect 4th factors.
= 4 16 x 8 6 x 3
Group the perfect 4th factors.
= 4 16 x 8 ⋅ 4 6 x 3
Factor into two radicals.
= 2 x 2 ⋅ 4 6 x3
Simplify the first radical.
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Objective #2: Example
2a. Simplify by factoring:
3
40
=
3
8⋅5
=
3
8⋅ 3 5
3
40
= 23 5
2b. Simplify by factoring:
200
=
x2 y
200x 2 y
100 x 2 ⋅ 2 y
= 10 x 2 y
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Objective #2: Example
2a. Simplify by factoring:
3
40
=
3
8⋅5
=
3
8⋅ 3 5
3
40
= 23 5
2b. Simplify by factoring:
200
=
x2 y
200x 2 y
100 x 2 ⋅ 2 y
= 10 x 2 y
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Objective #2: Example
2c. Express the function f ( x)=
f ( x)=
3 x 2 − 12 x + 12
=
3( x 2 − 4 x + 4)
=
3 x 2 − 12 x + 12 in simplified form.
3( x − 2) 2
= 3 ⋅ ( x − 2) 2
=
3⋅ x − 2
f ( x)= x − 2 3
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Objective #2: Example
2c. Express the function f ( x)=
f ( x)=
3 x 2 − 12 x + 12
=
3( x 2 − 4 x + 4)
=
3 x 2 − 12 x + 12 in simplified form.
3( x − 2) 2
= 3 ⋅ ( x − 2) 2
=
3⋅ x − 2
f ( x)= x − 2 3
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Simplifying Radicals
For Examples 2d and 2e and the remainder of this chapter, in
situations that do not involve functions, we will assume that no
radicands involve negative quantities raised to even powers.
Based upon this assumption, absolute value bars are not
necessary when taking even roots.
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Objective #2: Example
x9 y11z 3
2d. Simplify:
=
x9 y11z 3
x8 y10 z 2 ⋅ xyz
= x 4 y 5 z xyz
2e. Simplify:
3
3
40 x10 y14 =
40x10 y14
3
8 ⋅ 5 ⋅ x9 ⋅ x ⋅ y12 ⋅ y 2
= 3 8 x9 y12 3 5 xy 2
= 2 x3 y 4 3 5 xy 2
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Objective #2: Example
x9 y11z 3
2d. Simplify:
=
x9 y11z 3
x8 y10 z 2 ⋅ xyz
= x 4 y 5 z xyz
2e. Simplify:
3
3
40 x10 y14 =
40x10 y14
3
8 ⋅ 5 ⋅ x9 ⋅ x ⋅ y12 ⋅ y 2
= 3 8 x9 y12 3 5 xy 2
= 2 x3 y 4 3 5 xy 2
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Objective #3
Multiply radicals and then simplify.
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Multiplying Radicals
EXAMPLE
Multiply and simplify:
(a) 4 4 x 2 y 3 z 3 ⋅ 4 8 x 4 yz 6
(b) 5 8 x 4 y 3 z 3 ⋅ 5 8 xy 9 z 8 .
SOLUTION
(a) 4 4 x 2 y 3 z 3 ⋅ 4 8 x 4 yz 6
= 4 4 x 2 y 3 z 3 ⋅ 8 x 4 yz 6
Use the product rule.
= 4 32 x 6 y 4 z 9
Multiply.
= 4 16 ⋅ 2 x 4 x 2 y 4 z 8 z
Identify perfect 4th factors.
(
)(
= 4 16 x 4 y 4 z 8 2 x 2 z
)
= 4 16 x 4 y 4 z 8 ⋅ 4 2 x 2 z
Group the perfect 4th factors.
Factor into two radicals.
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Multiplying Radicals
CONTINUED
= 2 xyz 2 4 2 x 2 z
Factor into two radicals.
(b) 5 8 x 4 y 3 z 3 ⋅ 5 8 xy 9 z 8
= 5 8 x 4 y 3 z 3 ⋅ 8 xy 9 z 8
Use the product rule.
= 5 64 x 5 y12 z11
Multiply.
= 5 32 ⋅ 2 x 5 y10 y 2 z10 z
Identify perfect 5th factors.
(
)(
= 5 32 x 5 y10 z10 2 y 2 z
)
Group the perfect 5th factors.
= 5 32 x 5 y10 z10 ⋅ 5 2 y 2 z
Factor into two radicals.
= 2 xy 2 z 2 5 2 y 2 z
Simplify the first radical.
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Objective #3: Example
3a. Multiply and simplify:
6⋅ 2
6⋅ 2 =
12
=
4⋅3
=
4⋅ 3
=2 3
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Objective #3: Example
3a. Multiply and simplify:
6⋅ 2
6⋅ 2 =
12
=
4⋅3
=
4⋅ 3
=2 3
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Objective #3: Example
3b. Multiply and simplify: 10 3 16 ⋅ 5 3 2
10 3 16 ⋅ 5 3 2= 50 3 16 ⋅ 2
= 50 3 32
= 50 3 8 ⋅ 4
= 50 3 8 ⋅ 3 4
= 50 ⋅ 2 ⋅ 3 4
= 100 3 4
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Objective #3: Example
3b. Multiply and simplify: 10 3 16 ⋅ 5 3 2
10 3 16 ⋅ 5 3 2= 50 3 16 ⋅ 2
= 50 3 32
= 50 3 8 ⋅ 4
= 50 3 8 ⋅ 3 4
= 50 ⋅ 2 ⋅ 3 4
= 100 3 4
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Objective #3: Example
3c. Multiply and simplify:
4
4
4 x 2 y ⋅ 4 8 x6 y3
4
4 x 2 y ⋅ 4 8 x6 y3 =
32 x8 y 4
=
4
16 x8 y 4 ⋅ 4 2
= 2 x2 y 4 2
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Objective #3: Example
3c. Multiply and simplify:
4
4
4 x 2 y ⋅ 4 8 x6 y3
4
4 x 2 y ⋅ 4 8 x6 y3 =
32 x8 y 4
=
4
16 x8 y 4 ⋅ 4 2
= 2 x2 y 4 2
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